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What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?

Remarks: The same thing wouldn't be possible in $R^{2}$ by the Gauss-Bonnet theorem.
In 3 or more dimensions it is possible by a theorem of Lohkamp stated here (Theorem 1.6).

The same problem, with positive instead of negative curvature, has no solution by the Positive Energy Theorem (Theorem 1.2 in the link above).

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?

Basically, I am asking for a deformation of $R^{3}$ which decreases curvature.

Remarks: Such a thing wouldn't be possible in $R^{2}$ by the Gauss-Bonnet theorem.
In 3 or more dimensions it is possible by a theorem of Lohkamp stated here (Theorem 1.6).

The same problem, with positive instead of negative curvature, has no solution by the Positive Energy Theorem (Theorem 1.2 in the link above).

What's a simple example of metric on $R^{3}$ which has negative curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?

Remarks: The same thing wouldn't be possible in $R^{2}$ by the Gauss-Bonnet theorem.
In 3 or more dimensions it is possible by a theorem of Lohkamp stated here (Theorem 1.6).

The same problem, with positive instead of negative curvature, has no solution by the Positive Energy Theorem (Theorem 1.2 in the link above).

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?

Remarks: The same thing wouldn't be possible in $R^{2}$ by the Gauss-Bonnet theorem.
In 3 or more dimensions it is possible by a theorem of Lohkamp stated here (Theorem 1.6).

The same problem, with positive instead of negative curvature, has no solution by the Positive Energy Theorem (Theorem 1.2 in the link above).

What's a simple example of metric on $R^{3}$ which has negative curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?

Remarks: The same thing wouldn't be possible in $R^{2}$ by the Gauss-Bonnet theorem.
In 3 or more dimensions it is possible by a theorem of Lohkamp stated here.

The same problem, with positive instead of negative curvature, has no solution by the Positive Energy Theorem.

What's a simple example of metric on $R^{3}$ which has negative curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?

Remarks: The same thing wouldn't be possible in $R^{2}$ by the Gauss-Bonnet theorem.
In 3 or more dimensions it is possible by a theorem of Lohkamp stated here (Theorem 1.6).

The same problem, with positive instead of negative curvature, has no solution by the Positive Energy Theorem (Theorem 1.2 in the link above).